dihedral group: meaning, definition, pronunciation and examples
C2Formal/Technical
Quick answer
What does “dihedral group” mean?
In mathematics, a group of symmetries for a regular polygon, including rotations and reflections.
Audio
Pronunciation
Definition
Meaning and Definition
In mathematics, a group of symmetries for a regular polygon, including rotations and reflections.
In abstract algebra, any group generated by two elements, usually subject to specific relations (like r^n = e, s^2 = e, and srs = r^{-1}), which captures the structure of symmetries of an n-sided polygon. More broadly, it can refer to the symmetry group of any two-sided figure or prism.
Dialectal Variation
British vs American Usage
Differences
No lexical differences; the mathematical concept is identical. Spelling of related terms may follow regional conventions (e.g., 'symmetrise' vs. 'symmetrize').
Connotations
None beyond the technical mathematical meaning.
Frequency
Exclusively used in academic and professional mathematical contexts with equal frequency in both varieties.
Grammar
How to Use “dihedral group” in a Sentence
The dihedral group [of order 2n]A dihedral group [acts on a polygon][Rotations and reflections] generate a dihedral group.Vocabulary
Collocations
Examples
Examples of “dihedral group” in a Sentence
verb
British English
- The set dihedralises the polygon's symmetries. (Note: highly non-standard, constructed for example)
American English
- The transformations dihedralize the figure's symmetry. (Note: highly non-standard, constructed for example)
adjective
British English
- The dihedral symmetry of the snowflake was analysed.
American English
- The molecule's dihedral symmetry group is D3.
Usage
Meaning in Context
Business
Not used.
Academic
Core term in undergraduate abstract algebra, geometry, and crystallography courses.
Everyday
Not used.
Technical
Fundamental in mathematical research, computer graphics (for symmetry operations), and physics (particularly crystallography and molecular symmetry).
Vocabulary
Synonyms of “dihedral group”
Strong
Neutral
Weak
Vocabulary
Antonyms of “dihedral group”
Watch out
Common Mistakes When Using “dihedral group”
- Confusing the dihedral group D_n with the cyclic group C_n (which only has rotations).
- Misidentifying the order (D_n has order 2n, not n).
- Using 'dihedral' as an adjective for non-mathematical angles in place of 'dihedral angle'.
FAQ
Frequently Asked Questions
No, for n > 2, the dihedral group D_n is non-abelian because reflections and rotations do not generally commute.
It is typically generated by a rotation (r) and a reflection (s), with relations r^n = e, s^2 = e, and srs = r^{-1}.
It appears in crystallography (symmetry of certain crystals), computer graphics (object manipulation), and chemistry (symmetry of some molecules).
They often refer to the same group: the symmetry group of a line (or an infinite strip), generated by a translation and a reflection.
In mathematics, a group of symmetries for a regular polygon, including rotations and reflections.
Dihedral group is usually formal/technical in register.
Dihedral group: in British English it is pronounced /daɪˈhiːdrəl ɡruːp/, and in American English it is pronounced /daɪˈhidrəl ɡrup/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “None”
Learning
Memory Aids
Mnemonic
Imagine a DIamond (a regular polygon) being HELD and ROTated and flipped—its symmetry operations form the DIHEDRAL group.
Conceptual Metaphor
THE SYMMETRY OF A REGULAR SHAPE AS A SET OF ACTIONS (A dance of rotations and mirror flips).
Practice
Quiz
What is the order of the dihedral group D_n?